Concept

Stochastic block model

Summary
The stochastic block model is a generative model for random graphs. This model tends to produce graphs containing communities, subsets of nodes characterized by being connected with one another with particular edge densities. For example, edges may be more common within communities than between communities. Its mathematical formulation has been firstly introduced in 1983 in the field of social network by Paul W. Holland et al. The stochastic block model is important in statistics, machine learning, and network science, where it serves as a useful benchmark for the task of recovering community structure in graph data. Definition
The stochastic block model takes the following parameters:
  • The number n of vertices;
  • a partition of the vertex set {1,\ldots,n} into disjoint subsets C_1,\ldots,C_r, called communities;
  • a symmetric r \times r matrix P of edge probabilities.
The edge set is
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